Representational Decomposability

• Representational decomposability is the property of factorizing a system’s internal representations into interpretable, reusable units via additive, geometric, or structural methods.
• It enables applications such as improved few-shot learning, enhanced robustness, and more effective symbolic grounding in vision and language tasks.
• Empirical studies demonstrate decomposability through techniques like additive attribute embeddings, tangent-space approximations, and semantic-preserving subnetworks to align models with task-relevant structure.

Representational decomposability is the property that a representation can be factored into interpretable, reusable components rather than treated as a single opaque vector or monolithic function. In the recent literature, this idea appears in several technically distinct forms: additive attribute embeddings for recognition, symbolic grammars for abstract visual reasoning, part–prototype factorizations for images, geodesically additive tangent-space models for vision–language embeddings, and semantic-preserving subnetworks defined relative to a model’s decision boundary (Tokmakov et al., 2018, Vaishnav et al., 23 Apr 2026, Mishra et al., 2022, Berasi et al., 21 Mar 2025, Lee et al., 9 Apr 2026). Across these formulations, the common claim is not merely that a system can be partitioned, but that global behavior can be reconstructed or analyzed from structured local units.

1. Formal criteria and recurring dimensions

A first formalization treats decomposability as additive reconstruction from named primitives. In "Learning Compositional Representations for Few-Shot Recognition", each image $x$ is associated with a derivation $D(x)$ over an attribute vocabulary $\mathcal{D}_0$, and a representation $f_\theta(x)$ is compositional if

$$f_{\theta}(x_i) = \sum_{d \in D(x_i)} \hat{f}_{\eta}(d).$$

The paper then relaxes this hard equality to

$$f_\theta(x_i)= \sum_{d \in D(x_i)} \hat{f}_\eta(d) + w(x_i),$$

and adds an orthogonality penalty

$R_{\text{orth}}(\eta)= \mid \eta \eta^T - I \mid$

to encourage factorization of attribute directions (Tokmakov et al., 2018).

A second formalization is geometric rather than linear. "Not Only Text" defines geodesically decomposable embeddings by requiring primitive-specific tangent vectors $z_i^\# \in T_\mu M$ such that, for a composition $z=(z_1,\dots,z_s)$,

$u_{z} = \Exp_\mu \big( z_{1}^\# + \dots + z_{s}^\# \big),$

with the centering constraint

$D(x)$0

Here decomposability is not additive in ambient Euclidean space, but additive in the tangent space of a manifold such as the unit sphere $D(x)$1 (Berasi et al., 21 Mar 2025).

A third formalization is semantic and structural. "On the Decompositionality of Neural Networks" defines a decomposition

$D(x)$2

and requires boundary-aware semantic fidelity

$D(x)$3

together with structural divergence constraints

$D(x)$4

Decomposability is therefore defined as a semantic-preserving abstraction over a network’s internal supports, not as pruning alone (Lee et al., 9 Apr 2026).

A fourth formalization is explicitly hierarchical. In "Symbolic Grounding Reveals Representational Bottlenecks in Abstract Visual Reasoning", Bongard-LOGO is represented as

$D(x)$5

with images decomposed into LOGO-style primitives whose attributes are explicitly separable across type, style, and geometry (Vaishnav et al., 23 Apr 2026).

These definitions differ in ontology, but they recur along the same axes: a unit of decomposition, a composition rule, an explicit constraint against entanglement, and a criterion that the decomposition remain behaviorally meaningful. This suggests that representational decomposability is not a single formal object but a family of constraints on how structure is exposed.

2. Symbolic decomposability in abstract visual reasoning

In Bongard-LOGO, each problem contains six positive images $D(x)$6, six negative images $D(x)$7, and one query image $D(x)$8. The task is to infer a rule $D(x)$9 such that $\mathcal{D}_0$0, $\mathcal{D}_0$1, and then classify the query. The benchmark emphasizes geometric, relational, and topological concepts, and each image is generated by a ground-truth LOGO program. The paper’s Componential–Grammatical paradigm replaces pixels with symbolic inputs, either as Action Programs of the form line_TYPE_LENGTH-TURNANGLE and arc_TYPE_ARCANGLE_ARCRADIUS-TURNANGLE, or as Action Descriptions rendered as stepwise English clauses. Both formats preserve decomposition at the levels of problem, image, shape, and primitive action (Vaishnav et al., 23 Apr 2026).

This change in representation produces large empirical differences. The visual baseline, Gemini-2.5-Flash on pixels, reaches $\mathcal{D}_0$2 on Free-form, $\mathcal{D}_0$3 on Basic, and $\mathcal{D}_0$4 on Human-designed problems, essentially chance for a binary task. By contrast, pooled over 12 LLMs, C–G with Action Programs reaches $\mathcal{D}_0$5 on Free-form, $\mathcal{D}_0$6 on Basic, and $\mathcal{D}_0$7 on Human-designed, while Action Descriptions reach $\mathcal{D}_0$8, $\mathcal{D}_0$9, and $f_\theta(x)$0, respectively. The best model, Phi-4-Reasoning, attains $f_\theta(x)$1 on Free-form problems in the AP setting. Minimal-context prompting remains close to the full AP condition, with $f_\theta(x)$2 versus $f_\theta(x)$3 on Free-form, and concept conditioning adds only small pooled gains of $f_\theta(x)$4, $f_\theta(x)$5, and $f_\theta(x)$6 points on Free-form, Basic, and Human-designed splits. In a VLM-capable subset, grounded C–G with symbolic input plus the query image gives $f_\theta(x)$7, $f_\theta(x)$8, and $f_\theta(x)$9, compared with a visual-only baseline of $$f_{\theta}(x_i) = \sum_{d \in D(x_i)} \hat{f}_{\eta}(d).$$0, $$f_{\theta}(x_i) = \sum_{d \in D(x_i)} \hat{f}_{\eta}(d).$$1, and $$f_{\theta}(x_i) = \sum_{d \in D(x_i)} \hat{f}_{\eta}(d).$$2, so adding the query image does not systematically help (Vaishnav et al., 23 Apr 2026).

The significance of these numbers is narrow but strong. The paper explicitly treats symbolic grounding as a diagnostic probe, not a practical multimodal architecture. Its main claim is that the dominant bottleneck on Bongard-LOGO lies in representation rather than reasoning: when the input is already hierarchical, compositional, and factorized, the same class of LLMs can perform strong abstract visual reasoning.

3. Attribute, part, and primitive factorization in vision

In supervised recognition, decomposability has been enforced by attribute-aligned feature spaces. "Learning Compositional Representations for Few-Shot Recognition" uses category-level attribute annotations and a soft compositionality loss that aligns image features with positive attributes and pushes them away from negative ones. On CUB-200-2011 with a ResNet-10 and cosine classifier, 1-shot top-5 accuracy on novel classes rises from $$f_{\theta}(x_i) = \sum_{d \in D(x_i)} \hat{f}_{\eta}(d).$$3 to $$f_{\theta}(x_i) = \sum_{d \in D(x_i)} \hat{f}_{\eta}(d).$$4, and with data augmentation to $$f_{\theta}(x_i) = \sum_{d \in D(x_i)} \hat{f}_{\eta}(d).$$5. On SUN, it rises from $$f_{\theta}(x_i) = \sum_{d \in D(x_i)} \hat{f}_{\eta}(d).$$6 to $$f_{\theta}(x_i) = \sum_{d \in D(x_i)} \hat{f}_{\eta}(d).$$7, and with augmentation to $$f_{\theta}(x_i) = \sum_{d \in D(x_i)} \hat{f}_{\eta}(d).$$8. On ImageNet 1-shot novel classes, the method reaches $$f_{\theta}(x_i) = \sum_{d \in D(x_i)} \hat{f}_{\eta}(d).$$9, compared with a best previous $$f_\theta(x_i)= \sum_{d \in D(x_i)} \hat{f}_\eta(d) + w(x_i),$$0. Network Dissection analysis on SUN shows the number of interpretable units in the last layer increasing from 169 to 333, and unique concepts from 92 to 119 (Tokmakov et al., 2018).

A different construction appears in Recognition as Part Composition. RPC first extracts $$f_\theta(x_i)= \sum_{d \in D(x_i)} \hat{f}_\eta(d) + w(x_i),$$1 part-level features with a Multi-Attention CNN, then represents each part as a mixture over $$f_\theta(x_i)= \sum_{d \in D(x_i)} \hat{f}_\eta(d) + w(x_i),$$2 prototypes. For part $$f_\theta(x_i)= \sum_{d \in D(x_i)} \hat{f}_\eta(d) + w(x_i),$$3,

$$f_\theta(x_i)= \sum_{d \in D(x_i)} \hat{f}_\eta(d) + w(x_i),$$4

and the full image code is $$f_\theta(x_i)= \sum_{d \in D(x_i)} \hat{f}_\eta(d) + w(x_i),$$5. On miniImageNet, RPC reports $$f_\theta(x_i)= \sum_{d \in D(x_i)} \hat{f}_\eta(d) + w(x_i),$$6 1-shot and $$f_\theta(x_i)= \sum_{d \in D(x_i)} \hat{f}_\eta(d) + w(x_i),$$7 5-shot accuracy; on CUB few-shot, $$f_\theta(x_i)= \sum_{d \in D(x_i)} \hat{f}_\eta(d) + w(x_i),$$8 and $$f_\theta(x_i)= \sum_{d \in D(x_i)} \hat{f}_\eta(d) + w(x_i),$$9. In human evaluation, part-only class identification reaches $R_{\text{orth}}(\eta)= \mid \eta \eta^T - I \mid$0, and prototype recognition accuracies are $R_{\text{orth}}(\eta)= \mid \eta \eta^T - I \mid$1, $R_{\text{orth}}(\eta)= \mid \eta \eta^T - I \mid$2, and $R_{\text{orth}}(\eta)= \mid \eta \eta^T - I \mid$3 across the three parts. Under FGSM at $R_{\text{orth}}(\eta)= \mid \eta \eta^T - I \mid$4, RPC retains about $R_{\text{orth}}(\eta)= \mid \eta \eta^T - I \mid$5 accuracy on CUB and about $R_{\text{orth}}(\eta)= \mid \eta \eta^T - I \mid$6 on Cars, whereas the baselines drop below $R_{\text{orth}}(\eta)= \mid \eta \eta^T - I \mid$7 and about $R_{\text{orth}}(\eta)= \mid \eta \eta^T - I \mid$8, respectively (Mishra et al., 2022).

Physical Primitive Decomposition pushes factorization into geometry and physics. Each primitive is a rigid oriented box with size $R_{\text{orth}}(\eta)= \mid \eta \eta^T - I \mid$9, translation $z_i^\# \in T_\mu M$0, rotation $z_i^\# \in T_\mu M$1, and density $z_i^\# \in T_\mu M$2. The object representation is the set $z_i^\# \in T_\mu M$3, trained jointly from voxelized shape, RGB appearance, and physical trajectories. On block towers, geometric decomposition reaches F1 $z_i^\# \in T_\mu M$4; on tools, F1 $z_i^\# \in T_\mu M$5. For density estimation on tools, PPD full reports Top-1 $z_i^\# \in T_\mu M$6 and RMSE $z_i^\# \in T_\mu M$7, improving to $z_i^\# \in T_\mu M$8 and $z_i^\# \in T_\mu M$9 with sampling, compared with $z=(z_1,\dots,z_s)$0/$z=(z_1,\dots,z_s)$1 for image-only and $z=(z_1,\dots,z_s)$2/$z=(z_1,\dots,z_s)$3 for physics-only. In a real 2-block tower task, model-vs-truth correlation is $z=(z_1,\dots,z_s)$4, human-vs-truth $z=(z_1,\dots,z_s)$5, and human-vs-model $z=(z_1,\dots,z_s)$6 (Liu et al., 2018).

Taken together, these studies instantiate decomposability as explicit factorization into attributes, parts, prototypes, or physical primitives. They also show that the benefits are not identical: some gains appear as few-shot sample efficiency, some as interpretability, some as robustness, and some as better alignment with physically meaningful behavior.

4. Geometric and Jacobian-based decompositions

A post-hoc geometric account is given by Geodesically Decomposable Embeddings. For normalized VLM embeddings on the sphere, the paper maps image representations into a tangent space at the intrinsic mean $z=(z_1,\dots,z_s)$7, averages tangent vectors over slices that share a primitive, and reconstructs compositions through $z=(z_1,\dots,z_s)$8. On UT-Zappos closed-world compositional classification, CLIP zero-shot gives AUC $z=(z_1,\dots,z_s)$9, linear decomposition on image embeddings gives AUC $u_{z} = \Exp_\mu \big( z_{1}^\# + \dots + z_{s}^\# \big),$0, and GDE gives AUC $u_{z} = \Exp_\mu \big( z_{1}^\# + \dots + z_{s}^\# \big),$1 with HM $u_{z} = \Exp_\mu \big( z_{1}^\# + \dots + z_{s}^\# \big),$2. On MIT-States, the corresponding AUCs are $u_{z} = \Exp_\mu \big( z_{1}^\# + \dots + z_{s}^\# \big),$3, $u_{z} = \Exp_\mu \big( z_{1}^\# + \dots + z_{s}^\# \big),$4, and $u_{z} = \Exp_\mu \big( z_{1}^\# + \dots + z_{s}^\# \big),$5. In group robustness, GDE reaches worst-group accuracy $u_{z} = \Exp_\mu \big( z_{1}^\# + \dots + z_{s}^\# \big),$6 on Waterbirds and $u_{z} = \Exp_\mu \big( z_{1}^\# + \dots + z_{s}^\# \big),$7 on CelebA, with GAP $u_{z} = \Exp_\mu \big( z_{1}^\# + \dots + z_{s}^\# \big),$8 and $u_{z} = \Exp_\mu \big( z_{1}^\# + \dots + z_{s}^\# \big),$9, outperforming or matching task-specific debiasing baselines (Berasi et al., 21 Mar 2025).

A complementary argument is that global embedding geometry is not an adequate proxy for compositional structure. "Global Geometry Is Not Enough for Vision Representations" tests 21 vision encoders and finds near-zero correlation between compositional binding and standard geometry statistics: G.PR versus binding $D(x)$00, G.Iso $D(x)$01, and L.Iso $D(x)$02. In contrast, Jacobian Effective Rank correlates with binding at $D(x)$03, with cross-validated $D(x)$04; combining JER with the Same/Different structural control yields $D(x)$05 and leave-one-out $D(x)$06. The paper’s analytic account attributes this disparity to objective design: existing losses constrain embedding geometry but leave the local input–output mapping underconstrained (Chung et al., 3 Feb 2026).

A third geometric perspective models representations as coordinates on a random lattice with percolation structure. "Representation Learning on a Random Lattice" classifies learned features into context, component, and surface features. In the high-dimensional analysis, the lattice is approximated by a Bethe lattice with percolation threshold

$D(x)$07

and the model assigns high-level context features to clusters, component features to coordinates within a cluster, and surface features to format-specific or shortcut structure. The paper argues that cluster geometry has fractal dimension $D(x)$08, so many component features should be intrinsically multidimensional rather than one-dimensional linear directions (Brill, 28 Apr 2025).

These papers converge on a specific correction to a common simplification. A representation can be globally well-spread and still fail to expose the local factor structure needed for binding, robustness, or controlled composition. In that sense, decomposability is closer to geometry-aware factorization or functional sensitivity than to isotropy alone.

5. Distributional and linguistic decomposability

In idiom processing, decomposability has been recast as a model-internal property of contextualized sentence representations. "Rethinking the Idiomaticity Decomposability Hypothesis" defines decomposability as the degree of semantic alignment between an idiom sentence $D(x)$09 and a glossed sentence $D(x)$10, then measures token contribution by masking each idiom token $D(x)$11 and computing

$D(x)$12

Expression-level decomposability is then aggregated over $D(x)$13 by mean, maximum, Gini dispersion, or entropy. The paper evaluates BERT-base, BERT-large, ModernBERT, and OLMo checkpoints, and the best human-aligned configuration is BERT-large uncased, final layer, Wasserstein similarity, and sum aggregation, with correlation $D(x)$14 against human decomposability ratings (Mi et al., 2 Jun 2026).

The relationship to syntax is notably weak and often reversed. Across 527 idioms, the maximum reported decomposability–flexibility correlation is

$D(x)$15

and by idiom type the PP subset yields $D(x)$16, while the VP subset shows $D(x)$17. Human decomposability ratings do not significantly correlate with corpus-based syntactic flexibility. In pretraining analyses over 100 OLMo checkpoints, representational stabilization is not explained by frequency alone: surprisal, decomposability, and frequency all contribute, with interaction coefficients $D(x)$18, $D(x)$19, and $D(x)$20 for steps $D(x)$21 frequency, steps $D(x)$22 surprisal, and steps $D(x)$23 decomposability, respectively, and decomposability shows the strongest training-dependent effect (Mi et al., 2 Jun 2026).

This literature narrows the meaning of decomposability in language. It is not simply a proxy for human semantic transparency, nor does it robustly predict syntactic flexibility. Instead, in these results it behaves as an emergent distributional property of how figurative meaning is distributed across constituents during learning.

6. Semantic-preserving and representation-independent decomposition

Neural decompositionality makes decomposability into an explicit semantic contract. SAVED constructs component subnetworks $D(x)$24, mines low-margin inputs near the decision boundary, and learns structure-aware masks while tracking empirical disagreement

$D(x)$25

On BERT-small for DBPedia-14, LBMask reports $D(x)$26, Hoeffding-corrected $D(x)$27, max overlap $D(x)$28, min prune $D(x)$29, and boundary confusion deviation $D(x)$30, so the local contract is satisfied. On AG News, semantic fidelity fails despite acceptable structural statistics, and for ResNet-34 and DeiT-small on CIFAR-10, structured masking yields structural separation but boundary-local disagreement remains too high. The paper therefore reports a marked architectural difference: BERT on DBPedia-14 is decompositional under the proposed contract, whereas the tested vision models are not (Lee et al., 9 Apr 2026).

A more abstract formalization appears in "Representation Independent Decompositions of Computation". The paper generalizes from semigroups to semigroupoids and gives a decomposition procedure with three stages: collapse, copy, and compress. Starting from a surjective relational functor $D(x)$31, it defines the tracing product $D(x)$32, compresses equivalent preimages into a kernel $D(x)$33, and proves that the pinhole cascade product $D(x)$34 emulates the original system: $D(x)$35 The point of the construction is that the decomposition is stated in terms of arrows, objects, and relational functors rather than any particular state encoding, so decomposability is representation independent in a categorical sense (Egri-Nagy et al., 7 Apr 2025).

These works move decomposability away from heuristic modularization. In one case the criterion is preservation of decision-boundary semantics under structural separation; in the other it is an iterative algebraic factorization of computation that is independent of concrete representation.

7. Scope, misconceptions, and open questions

Several recurring caveats limit how far current claims can be taken. In Bongard-LOGO, symbolic grounding is explicitly a diagnostic upper bound with oracle access to ground-truth programs rather than a learned perception system, and even with perfect symbolic input, Human-designed accuracy remains around $D(x)$36, so residual reasoning difficulty remains (Vaishnav et al., 23 Apr 2026). In attribute-based compositional representation learning, hard equality $D(x)$37 is often too strong because attributes are not exhaustive; the practical method is the softer additive-plus-residual form, and the orthogonality constraint may conflict with correlated attributes (Tokmakov et al., 2018). In GDE, decomposability depends on a predefined factorization such as attribute $D(x)$38 object and on a tangent-space approximation that is justified only locally; the method does not discover factorization unsupervised (Berasi et al., 21 Mar 2025). In vision representation analysis, global isotropy and participation ratio are shown to be nearly blind to compositional binding, so geometry alone should not be treated as a proxy for decomposability (Chung et al., 3 Feb 2026). In language, model-derived idiom decomposability correlates only weakly with human judgments and shows a small negative relationship with syntactic flexibility, which cautions against equating representational decomposability with traditional psycholinguistic decomposability (Mi et al., 2 Jun 2026). In neural decompositionality, pruning is not sufficient: unstructured pruning can preserve semantics while collapsing structure, and structured pruning can preserve structure while violating boundary semantics (Lee et al., 9 Apr 2026).

A recurring misconception is that decomposability is synonymous with interpretability or sparsity. The cited work does not support that equivalence. In these papers, decomposability is variously tied to additive recoverability, grammar-like structure, manifold-aware composition, local functional sensitivity, or semantic-preserving modularity. A plausible implication is that decomposability is task-aligned rather than universal: representations appear decomposable when their factorization matches the causal, relational, or decision-boundary structure that the task actually requires.